A bit introduction to foliation in GR and some question.

[Kevin_spencer2]

If we have Einstein-Hilbert Lagrangian so:

$$\mathcal L = (-g)^{1/2}R$$ for R Ricci scalar (¿?) then my question perhaps mentioned before is if we can split the metric into:

$$g_{ab}= Ndt^{2}+g_{ij} dx^{i} dx^{j}$$ N=N(t) 'lapse function'

then i would like to know if somehow you can split the Lagrangian into:

$$\mathcal L = (-g)^{1/2}R=N^{1/2}(g^{3})^{1/2}g^{00}R_{00}+N^{1/2}(g^{3})^{1/2}g^{ij}R_{ij}$$


Following Einstein equations then $$R_{00}=T_{00}= \rho$$ 'energy density equation' and

$$\iiint (g^{3})^{1/2}R^{(3)}d^{3}x = 2T$$

$$\iint (g^{(3)})^{1/2}R_{00}=\mathcal H = E$$

Is some kind of Kinetic energy so the Einstein-Hilbert Lagrangian takes the form:

$$\mathcal L =\int_{a}^{b} dt(T-V)$$ a Kinetic plus potential term.

 

[Chris Hillman]

For one thing, the Ricci tensor computed from the (three-dimensional) intrinsic geometry of the hyperslice is not the same as the restriction of the Ricci tensor (computed for the spacetime itself) to said hyperslice.